Planty of stuff in Italy either in Piemonte or Liguria...but you need to do some research I am not an expert yet...
I am assuming you are talking about hiking...
In Chamonix without crampons you are limited to that...but you can rent boots, crampons and ice axe in town and go for some nice solo climbing on the high mountains...
But if you do not want want to rent gear your option is the Aiguille Rouge. There is a beautiful and engaging traverse that you can do from Le Brevent all the way to Lac Blanc and further, it involves some scrambling, ladder climbing, snowfield crossing and a lot of up and down. I think the time required is about 8 hrs, but I never done it...

Can't tell you much about the Italian side of Mont Blanc though I'd imagine there's plenty to do spanning from the Petit Saint Bernard Col to the Grand Saint Bernard. If you're driving there's a few options around and above Emosson/Susanfe/Barme. You could try les Dents du Midi, you'd be hard pushed to find a better view without gear or a cable car than the Haute Cime but it depends on conditions at the time. Mont Ruan too. The obvious choice for a 3000er around Chamonix is Mont Buet. Choices depend also on whether you are willing to do stay overnight at a refuge or bivi. Buet is doable in a day with an early start. Mont Tondu (above the Tre la Tete glacier) can also be done with very little gear via the Lacs Jovet., just a piolet is advisable (I don't know the current conditions up there though) and it is long. There's also a number of worthy hikes and scrambles just below the 3,000mark, such as Pic de Tenneverge Par le Pas Noir - which is amazing, difficult and remote but very treacherous in wet weather. Closer to chamonix there is also Aiguille de Genepi and Pointes des Grandes above the Refuge Albert 1er.
http://www.summitpost.org/view_object.p ... _id=609471

i know it's not quite chamonix, but i'll be a bit further west in Swizterland (arolla) this week - until the 11th, with some climbing gear but no one to climb with. there are plenty of good peaks there, if by any chance you'd like to head over and meet up let me know.
jon